Approximations for nonlinear mappings by the hybrid method in Hilbert spaces

作者：Nakajo Kazuhide^*; Shimoji Kazuya; Takahashi Wataru

来源：Nonlinear Analysis-Theory Methods & Applications, 2011, 74(18): 7025-7032.

DOI：10.1016/j.na.2011.07.023

• Strong convergence
• Hybrid method
• Asymptotically kappa-strict pseudo-contractions
• Lipschitz and pseudo-contractive mappings
• One-parameter uniformly Lipschitz semigroups of pseudo-contractive mappings

The hybrid method in mathematical programming was introduced by Haugazeau (1968) [1] and he proved a strong convergence theorem for finding a common element of finite nonempty closed convex subsets of a real Hilbert space. Later, Bauschke and Combettes (2001) [2] proposed some condition for a family of mappings (the so-called coherent condition) and established interesting results by the hybrid method. The authors (Nakajo et al., 2009) [10] extended Bauschke and Combettes's results. In this paper, we introduce a condition weaker than the coherent condition and prove strong convergence theorems which generalize the results of Nakajo et al. (2009) [10]. And we get strong convergence theorems for a family of asymptotically.-strict pseudo-contractions, a family of Lipschitz and pseudo-contractive mappings and a one-parameter uniformly Lipschitz semigroup of pseudo-contractive mappings.

• 出版日期2011-12